Fitting GP to lightcurve 428_…in Stan
Notebook outlining the fitting of GP to thunderKAT lightcurve ID$ 428_…
1 Light Curve
- The light curve has \(N =\) 21 observations over a range of 147.57 days.
- Observations are evenly spread over the time range.
- The shortest gap between observations is 5 days.
- The longest gap between observations is 14.68 days.
- The mean flux density is \(\bar{y} =\) 0.19 Jy.
- The mean standard error in the observations is 0.000491 Jy.
- The observational noise is very small relative to the brightness of the observations.
2 SE Basic Model
- Zero constant mean function.
- Squared Exponential kernel function.
- Homoskedastic noise.
- Wide prior on observational noise, uninformed by observational noise estimates.
\[y \sim \mathcal{N}(f(x), \sigma_\textrm{noise}^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_\textrm{noise} \sim \mathcal{N}^+(0,1)\]
2.1 MCMC Results
Warning: 2 of 4000 (0.0%) transitions ended with a divergence.
See https://mc-stan.org/misc/warnings for details.$num_divergent
[1] 2 0 0 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 1.043763 1.221051 1.102958 1.218542 variable mean median sd mad q5 q95 rhat ess_bulk
eta 0.0960 0.0421 0.1648 0.0386 0.0092 0.3781 1.0007 2447
ell 162.1374 139.9640 90.2990 67.1896 66.0926 330.1677 1.0013 2901
sigma 0.0063 0.0061 0.0012 0.0011 0.0047 0.0085 1.0000 3299
ess_tail
2595
2530
23102.2 MCMC Plots
2.3 Posterior Predictive Samples
The fitted model has a very long lengthscale, comparable to the length of the observational window. The estimated observational noise has a standard deviation more than an order of magnitude of that in the original data. The combination of these parameters has lead to a very smooth fit that passes through the middle of the observed data points rather than through any datapoints themselves.
2.4 PSD
3 SE Observational Errors Model
- Zero constant mean function.
- Squared exponential kernel function.
- Heteroskedastic (Gaussian) noise.
- Incorporate data on error in observations of each \(y_i\).
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
$num_divergent
[1] 0 0 0 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 0.9342605 1.0233746 0.9841971 1.05953533.1 MCMC Results
variable mean median sd mad q5 q95 rhat ess_bulk
eta 0.02216 0.01994 0.01039 0.00778 0.01081 0.04093 1.00042 6217
ell 11.22759 11.23455 0.51423 0.49400 10.35169 12.04060 1.00085 6382
sigma[1] 0.00039 0.00039 0.00004 0.00004 0.00033 0.00046 1.00137 8109
ess_tail
2839
2855
28663.2 MCMC Plots
3.3 Posterior Predictive Samples
By including the observed observational errors for setting priors on the Gaussian noise of each observation, the fitted median passes through each of the observed points.
3.4 PSD
4 Matern 3/2 kernel
- Matern 3/2 covariance kernel
- Zero constant mean function
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
$num_divergent
[1] 0 0 0 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 1.023875 1.080972 1.061123 1.0479264.1 MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.04154 0.02482 0.06405 0.01692 0.00897 0.12613 1.00033
ell 86.05902 79.21425 30.28509 22.88986 52.29066 141.91955 1.00006
sigma[1] 0.00039 0.00039 0.00004 0.00004 0.00032 0.00046 1.00113
ess_bulk ess_tail
3175 2357
3121 2219
7029 30634.2 MCMC Plots
4.3 Posterior Predictive Samples
4.4 PSD
5 SE + Matern 3/2 additive kernel
- Sum of squared exponential and Matern 3/2 kernels
- single output scale (marginal variance) hyperparameter
- zero constant mean function
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \left[ \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} \right]\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
5.1 MCMC Results
$num_divergent
[1] 0 0 0 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 1.0384751 1.0393340 0.9615068 0.9812617 variable mean median sd mad q5 q95 rhat
eta 0.01194 0.00898 0.01151 0.00477 0.00394 0.02844 1.00093
ell_SE 40.27136 35.43965 20.98407 13.58655 19.76977 76.49121 1.00026
ell_M 55.55175 52.95730 15.53669 12.78883 35.85322 82.75450 0.99958
sigma[1] 0.00039 0.00039 0.00004 0.00004 0.00032 0.00046 1.00001
ess_bulk ess_tail
3476 2008
4438 2183
3255 2159
5398 27055.2 MCMC Plots
5.3 Posterior Predictive Samples
5.4 PSD
6 SE + Matern 3/2 (2 output scales) additive kernel
- Sum of squared exponential and Matern 3/2 kernels
- One output scale (marginal variance) hyperparameter for each kernel
- zero constant mean function
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta_\textrm{SE}^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \eta^2_\textrm{M}\left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\}\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta_\textrm{SE} \sim \mathcal{N}^+(0,1)\]
\[\eta_\textrm{M} \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
Warning: 5 of 8000 (0.0%) transitions ended with a divergence.
See https://mc-stan.org/misc/warnings for details.$num_divergent
[1] 3 0 1 1
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 1.0236362 0.9947824 1.0108117 0.96146246.1 MCMC Results
variable mean median sd mad q5 q95 rhat ess_bulk
eta_SE 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 1.0006 6224
eta_M 0.0357 0.0183 0.0692 0.0136 0.0057 0.1101 0.9998 8500
ell_SE 1.2993 1.0819 0.8226 0.4905 0.5507 2.7425 1.0003 9555
ell_M 317.2728 258.4510 228.5529 151.3097 97.7173 744.3742 1.0003 6149
sigma[1] 0.0004 0.0004 0.0000 0.0000 0.0003 0.0005 1.0002 13074
ess_tail
3197
5063
4418
5179
54946.2 MCMC Plots
6.3 Posterior Predictive Samples
6.4 PSD
7 SE x Matern 3/2 multiplicative kernel
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\}\left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\}\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
Warning: 72 of 8000 (1.0%) transitions ended with a divergence.
See https://mc-stan.org/misc/warnings for details.$num_divergent
[1] 0 0 72 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 0.9895645 0.9546933 0.8873692 1.01267447.1 MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.03333 0.02631 0.04699 0.01695 0.00984 0.07239 1.12222
ell_SE 41.47464 41.01155 33.60317 25.98761 0.73175 97.00669 1.52568
ell_M 57.84199 65.94690 38.30645 26.86249 0.73907 110.92935 1.52388
sigma[1] 0.00039 0.00039 0.00004 0.00004 0.00033 0.00046 1.00042
f_star[1] 0.18392 0.18392 0.00040 0.00039 0.18326 0.18458 1.00020
ess_bulk ess_tail
22 2881
7 27
7 28
7949 5107
7611 78137.2 MCMC Plots
7.3 Posterior Predictive Samples
7.4 PSD
8 SE + Periodic
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \left[ \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \exp\left\{ -\frac{2 \sin^2\left( \pi\frac{\sqrt{(x - x')^2}}{T}\right)}{\ell_\mathrm{P}^2}\right\} \right]\]
\[\ell_\mathrm{P} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[T \sim \mathcal{U}[\textrm{minimum gap in x}, \textrm{range of x}]\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
8.1 MCMC Results
Warning: 26 of 8000 (0.0%) transitions ended with a divergence.
See https://mc-stan.org/misc/warnings for details.$num_divergent
[1] 1 12 3 10
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 0.9677225 1.0269724 0.9582524 1.0597445 variable mean median sd mad q5 q95 rhat ess_bulk
eta 0.0036 0.0032 0.0018 0.0013 0.0017 0.0068 1.0026 2771
ell_SE 9.9347 9.9336 0.7520 0.6500 8.7242 11.1683 1.0043 581
ell_P 2.7280 2.4112 1.3876 0.9705 1.2502 5.2363 1.0046 1307
T 92.5493 95.0402 34.0438 40.6806 36.5269 142.2803 1.0023 1305
ess_tail
2963
174
257
5138.2 MCMC Plots
8.3 Posterior Predictive Samples
8.4 PSD
9 SE + Matern 3/2 + Periodic
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta \left[ \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} + \exp\left\{ -\frac{2 \sin^2\left( \pi\frac{\sqrt{(x - x')^2}}{T}\right)}{\ell_\mathrm{P}^2}\right\} \right]\]
\[\ell_\mathrm{P} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[T \sim \mathcal{U}[\textrm{minimum gap in x}, \textrm{range of x}]\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
9.1 MCMC Results
Warning: 7 of 8000 (0.0%) transitions ended with a divergence.
See https://mc-stan.org/misc/warnings for details.$num_divergent
[1] 7 0 0 0
$num_max_treedepth
[1] 0 0 0 0
$ebfmi
[1] 0.8965104 0.9769791 0.9785691 0.9511891 variable mean median sd mad q5 q95 rhat ess_bulk
eta 0.0041 0.0033 0.0030 0.0016 0.0016 0.0089 1.0150 291
ell_SE 25.8580 23.3266 15.3052 8.9439 1.7806 50.8321 1.0716 37
ell_M 35.7989 36.0480 13.1031 9.5932 2.0152 55.2239 1.0716 38
ell_P 2.5098 2.2009 1.2929 0.8982 1.1765 4.8095 1.0035 3120
T 108.3781 114.5820 29.7569 30.1309 51.7676 144.4434 1.0137 300
ess_tail
2822
11
11
3895
1049.2 MCMC Plots
9.3 Posterior Predictive Samples
9.4 PSD
